What does "works" mean, because the result should be different between the two. The first example would seem to write the same value twice, so it's not as if the LEDs / GPIO state would change.

Check the GPIO configuration, and make sure all the bits you want set/clear actually get into the final state you want/expect, especially where there are multiple bits involved, and the initial state is unknown/unexpected. Use a debugger, review the registers.

"Works" means the LEDs (on PC8 and PC9) blink. My understanding is that BSRR will set specific bits, and BRR should clear them. I also found that I an clear the bits by using BSRR, but shifting the specific bits 16 over to the left, so "GPIOC->BSRR = 0x03000000;" will turn the LEDs off. But if I use BRR I should not need to shift the bits over 16 to the left. So why doesn't BRR work to turn the LEDs off as I have it?